Infinite Unicorn Paths and Gromov Boundaries

TL;DR

This paper extends unicorn paths to include geodesics asymptotic to laminations, providing new proofs for the identification of Gromov boundaries of curve and arc graphs as lamination spaces.

Contribution

It introduces a generalized notion of unicorn paths involving laminations and offers alternative proofs for known boundary identifications.

Findings

Gromov boundaries of curve and arc graphs are spaces of laminations.

Extended unicorn paths connect arcs and geodesics asymptotic to laminations.

New proofs simplify understanding of boundary structures.

Abstract

We extend the notion of unicorn paths between two arcs introduced by Hensel, Przytycki and Webb to the case where we replace one arc with a geodesic asymptotic to a lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the arc graph, respectively, as spaces of laminations.